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To find the determinant of the given \(3 \times 3\) matrix \(A\):

\[
|A| = \begin{vmatrix} 1 & -2 & 1 \\ 2 & 2 & -3 \\ 1 & 1 & -2 \end{vmatrix}
\]

we can use the cofactor expansion along the first row. The formula for the determinant of a \(3 \times 3\) matrix is:

\[
|A| = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

For the matrix above, assign the elements as follows:
\[
a = 1, \, b = -2, \, c = 1
\]
\[
d = 2, \, e = 2, \, f = -3
\]
\[
g = 1, \, h = 1, \, i = -2
\]

Now, calculate each term:

1. \( a(ei - fh) = 1 \times (2 \times -2 - (-3 \times 1)) = 1 \times (-4 + 3) = 1 \times -1 = -1 \)
2. \( -b(di - fg) = -(-2) \times (2 \times -2 - (-3 \times 1)) = 2 \times (-4 + 3) = 2 \times -1 = -2 \)
3. \( c(dh - eg) = 1 \times (2 \times 1 - 2 \times 1) = 1 \times (2 - 2) = 1 \times 0 = 0 \)

Adding these up:

\[
|A| = -1 - 2 + 0 = -3
\]

So, the determinant of the matrix \(A\) is \(-3\).

Would you like a detailed breakdown of each step or have further questions?

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Here are five related questions that might be of interest:

1. How is cofactor expansion used for larger matrices, like \(4 \times 4\)?
2. What properties of determinants can help simplify calculations?
3. How do you interpret the determinant in terms of transformations in geometry?
4. What does a zero determinant imply about a matrix?
5. How can you compute determinants quickly with row or column operations?

**Tip:** Remember, if any row or column in a matrix is a scalar multiple of another, the determinant will be zero, which can save time in calculations.

Learn how to find the determinant of a 3x3 matrix using the cofactor expansion method. This problem involves key concepts in linear algebra, including matrix operations and determinants, suitable for high school students in Grades 10-12.

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